Econometrics Analysis of Sales, Price and Advertising with OLS in Gretl

Posted on Oct 14, 2026 in Statistics

a) Econometrics Utility: Econometrics applies statistical and mathematical methods to quantify economic relationships, test theoretical models against real-world data, and forecast future trends.
b) Data Type and Range: The dataset consists of monthly time series data spanning from January 2015 to April 2024.
c) Variables: The endogenous (dependent) variable is the sales revenue (Sales). The explanatory (independent) variables are the product price (Price) and annual advertising expenditure (Advertising).
d) Random Disturbance Hypotheses: To estimate the model via Ordinary Least Squares (OLS), the errors must have a zero conditional mean, constant variance (homoscedasticity), no autocorrelation, and be normally distributed (for valid hypothesis testing).

Scatter Plot and Model Estimation

a) Scatter Plot: In Gretl, go to View > Graph specified vars > X-Y scatter, selecting Sales for the Y-axis and Price for the X-axis. This cloud of points will visually display whether the variables have a positive or negative linear relationship.
b) Population Model: The population regression model is $Sales_t = \beta_1 + \beta_2 Price_t + u_t$. The expected sign for the population parameter is negative, reflecting the standard economic law of demand (as price increases, sales volume decreases).
c) Model Estimation: In Gretl, navigate to Model > Ordinary Least Squares, selecting Sales as the dependent variable and Price as the independent variable. The Sample Regression Function (SRF) is written as follows. The estimated constant represents the theoretical average sales revenue if the product price were exactly €0.

a) Gauss-Markov Theorem: This theorem states that under the classical linear regression assumptions, the OLS estimators are the Best Linear Unbiased Estimators (BLUE), meaning they have the lowest variance among all linear, unbiased estimators.
b) Correlation and R-squared: In Gretl, calculate the correlation by selecting both variables and navigating to View > Correlation matrix. Squaring this correlation coefficient mathematically yields the univariate model’s coefficient of determination ($R^2$). This $R^2$ value represents the exact proportion of the total variation in sales that is explained by the price.

a) Slope Interval: In the Gretl estimation window, go to Analysis > Confidence intervals for coefficients. This provides the 95% confidence interval, indicating the range in which the true population slope lies with 95% probability.
b) Slope Significance: If the generated 95% confidence interval does not contain the value 0, the slope is statistically significant at the 5% level.
c) Model Significance: Test the global significance by examining the F-statistic and its associated critical region (p-value) at the bottom of the Gretl output. In a univariate model, the global F-test and the individual t-test for the slope are mathematically equivalent ($F = t^2$) and will yield the exact same conclusion regarding significance.

a) Re-estimation: In Gretl, go to Model > Ordinary Least Squares and add Advertising to the list of independent variables alongside Price.
b) Coefficient Interpretation: In this multivariate model, the estimated coefficient accompanying the price variable represents the expected change in average sales for a 1-euro increase in price, assuming advertising expenditure remains constant (ceteris paribus).

a) Elasticity Calculation: Obtain the variable means via View > Summary statistics. Calculate the average elasticity for both variables manually using the formula $E_i = \hat{\beta}_i \times (\bar{X}_i / \bar{Y})$. The variable that yields the highest absolute elasticity value has the greatest proportional effect on average sales volume.
b) Multivariate Significance: Test the significance of the new multivariate model by checking whether the F-statistic’s p-value in the estimation output is less than the 0.05 threshold.

a) Theoretical Effects: Converting the data by multiplying by the €1 = $1.1749 exchange rate represents a uniform change of scale for all monetary variables. Because the dependent variable (Sales) and explanatory variables (Price, Advertising) are multiplied by the exact same constant, the estimated slope parameters ($\hat{\beta}_2, \hat{\beta}_3$) will remain unchanged. The intercept and standard errors will scale by the 1.1749 factor.
b) Goodness-of-fit: The model’s goodness-of-fit ($R^2$) is scale-invariant and will remain identical to the model estimated in euros.